3A.11 Matrix Exponential

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The matrix exponential is given by the infinite series

expðAtÞ ¼ I þ At þ

1

2!

A2t2 þ · · · ð3A:26Þ

exactly like the scalar exponential

expðltÞ ¼ 1 þ lt þ

1

2!

l2t2 þ · · · ð3A:27Þ

The matrix exponential may be determined by reducing the infinite series given in Equation 3A.26

into a finite matrix polynomial of order n 2 1 (where, A is n £ n) by using the Cayley – Hamilton

theorem.

3A.11.1 Cayley – Hamilton Theorem

This theorem states that a matrix satisfies its own characteristic equation. The characteristic polynomial

of A can be expressed as

DðlÞ ¼ detðA 2 lIÞ ¼ anln þ an21ln21 þ · · · þ a0 ð3A:28Þ

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in which det( ) denotes determinant. The notation

DðAÞ ¼ anAn þ an21An21 þ · · · þ a0I ð3A:29Þ

is used. Then, by the Cayley – Hamilton theorem, we have

0 ¼ anAn þ an21An21 þ · · · þ a0I ð3A:30Þ

3A.11.2 Computation of Matrix Exponential

Using the Cayley – Hamilton theorem, we can obtain a finite polynomial expansion for expðAtÞ: First, we

express Equation 3A.26 and Equation 3A.27 as

expðAtÞ ¼ SðAÞ·DðAÞ þ an21An21 þ an22An22 þ · · · þ a0I ð3A:31Þ

expðltÞ ¼ SðlÞ·DðlÞ þ an21ln21 þ an22ln22 þ · · · þ a0 ð3A:32Þ

in which Sð·Þ is an appropriate infinite series, which is the result of dividing the exponential (infinite)

series by the characteristic polynomial Dð·Þ:

Next, since DðAÞ ¼ 0 by the Cayley – Hamilton theorem, Equation 3A.31 becomes

expðAtÞ ¼ an21An21 þ an22An22 þ · · · þ a0I ð3A:33Þ

Now it is just a matter of determining the coefficients, a0; a1; …; an21; which are functions of time. This

is done as follows. If l1; l2; …; ln are the eigenvalues of A; however, then, by definition

DðliÞ ¼ detðA 2 liIÞ ¼ 0 for i ¼ 1; 2; …; n ð3A:34Þ

Thus, from Equation 3A.32, we obtain

expðlitÞ ¼ an21ln21

i þ an22ln22

i þ · · · þ a0 for i ¼ 1; 2; …; n ð3A:35Þ

If the eigenvalues are all distinct, Equation 3A.35 represents a set of n independent algebraic equations

from which the n unknowns a0; a1; …; an21 could be determined. If some eigenvalues are repeated, the

derivatives of the corresponding equations (Equation 3A.35) have to be used as well.

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